Primary Decomposition

Definition

Primary decomposition is the process of expressing an ideal in a commutative ring as an intersection of primary ideals. This concept plays a crucial role in understanding the structure of ideals and their associated prime ideals, linking various important concepts such as localization, depth, and dimension in ring theory.

5 Must Know Facts For Your Next Test

1. Every ideal in a Noetherian ring can be expressed as an intersection of primary ideals, which leads to a unique primary decomposition up to reordering.
2. The primary components in a primary decomposition are directly related to the associated primes of the ideal, helping to analyze its algebraic properties.
3. Primary decomposition allows for simplifications when studying modules over rings, providing insights into their structure and properties.
4. In relation to localization, primary decomposition can be performed in local rings, simplifying calculations involving local properties of ideals.
5. The concept is crucial for understanding depth and dimension in Noetherian rings, as it helps clarify relationships between ideals and their generators.

Review Questions

• How does primary decomposition facilitate the analysis of associated primes in the context of an ideal's structure?
  • Primary decomposition allows us to break down an ideal into its primary components, each associated with specific prime ideals. By expressing an ideal as an intersection of primary ideals, we can identify the associated primes more easily. This understanding is vital because the associated primes reveal information about the ideal's generators and provide insight into its algebraic properties and behavior under localization.
• Discuss how primary decomposition interacts with localization in a commutative ring and its implications on the structure of ideals.
  • In a commutative ring, localization allows us to focus on specific elements or primes by creating a new ring where these elements become invertible. When applying primary decomposition within this localized context, we can refine our understanding of how ideals behave locally. This interaction shows how global properties can be influenced by local structures and highlights the importance of considering both aspects when analyzing ideals.
• Evaluate the significance of primary decomposition in the study of height and depth within Noetherian rings.
  • Primary decomposition plays a critical role in relating height and depth in Noetherian rings by providing insight into the chain lengths of prime ideals. By breaking down an ideal into primary components, we can assess how many steps it takes to reach certain prime ideals, thereby determining their heights. Additionally, this analysis informs our understanding of the depth of modules over these rings, illustrating how closely linked these concepts are through primary decompositions.

Related terms

Primary Ideal: An ideal I in a ring R is called primary if whenever ab ∈ I for some a, b ∈ R, then either a ∈ I or some power of b is in I.

Associated Prime: An associated prime of an ideal I is a prime ideal that corresponds to a minimal prime over I, reflecting the structure of the ring and the ideal.

Height of a Prime Ideal: The height of a prime ideal P is the length of the longest chain of prime ideals contained within P, which relates to the dimension of the ring.